A329170 Numbers of the form k^2 + 2 that are the sums of two squares.

2, 18, 146, 578, 1602, 2306, 3602, 4626, 5186, 7058, 10818, 12546, 17426, 19602, 20738, 21906, 30978, 32402, 36866, 41618, 46658, 48402, 63506, 69698, 76178, 80658, 97346, 102402, 104978, 132498, 138386, 147458, 153666, 156818, 166466, 176402, 183186, 197138

Offset

1,1

References

Gelca, R. and Andreescu, T. (2007). Putnam and beyond. New York: Springer. Page 278, Question 807.

Links

Table of n, a(n) for n=1..38.

Formula

The formula ((n^2 - 3)^2)/4 + n^2, where n is odd, generates the subsequence 2, 18, 146, 578, 1602, 3602, 7058, 12546, 20738, 32402, 48402 ... .

Example

1) k^2 + 2 = 0^2 + 2 = 2 = 1^2 + 1^2 = a^2 + b^2
2) k^2 + 2 = 4^2 + 2 = 18 = 3^2 + 3^2 = a^2 + b^2
3) k^2 + 2 = 12^2 + 2 = 146 = 5^2 + 11^2 = a^2 + b^2
4) k^2 + 2 = 24^2 + 2 = 578 = 7^2 + 23^2 = a^2 + b^2 (also, 578 = 17^2 + 17^2)
5) k^2 + 2 = 40^2 + 2 = 1602 = 9^2 + 39^2 = a^2 + b^2

Prog

(Python)
# A list of squares of integers. Currently contains 0 and 1; will be added elements below when calling 'procedure':
squaresList = [0, 1]
# We define a function that, given k**2+2, finds if it is a sum of two of the squares stored in the list:
def find_if_sum (a_var):
  for a_squared in range (0, len(squaresList)):
    for b_squared in range (a_squared, len(squaresList)):
      if (a_var == squaresList[a_squared] + squaresList[b_squared]):
        print (str(a_var) + ", " + "a_squared: " + str(squaresList[a_squared]) + " and b_squared: " + str(squaresList[b_squared]), sep = '', )
# We call this function on the base cases.
my_var = 0**2 + 2               # Here we directly compute k**2 + 2 for k = 0
find_if_sum (my_var)            # This is the call to the function
my_var = 1**2 + 2               # Here we directly compute k**2 + 2 for k = 1
find_if_sum (my_var)            # This is the call to the function
def procedure ():
  for k in range (2, 1501):      # Check all integers up to 1500, including
    my_var = k**2
    squaresList.append(my_var)
    my_var += 2                 # Add 2 to the square of k
    find_if_sum (my_var)        # This is the call to the function
procedure()                     # Call the procedure
(Magma) a:=[]; for m in [0..500] do for k in [0..m+1] do if IsSquare(m^2+2-k^2) then  Append(~a, m^2+2); break; end if; end for; end for; a; // Marius A. Burtea, Nov 07 2019

Crossrefs

Cf. A274595.

Intersection of A000404 and A059100.

Sequence in context: A290215 A208654 A037565 * A125835 A356623 A091170

Adjacent sequences: A329167 A329168 A329169 * A329171 A329172 A329173

Keyword

nonn

Author

Miriam Briskman, Nov 06 2019

Extensions

-

Status

approved